FlowSDE: a flow-matching-based SDE framework for predicting state transitions

Stochastic differential equations (SDEs) are effective tools for modelling various real-world phenomena, ranging from chemical reactions to neural dynamics. In this paper, we propose a flow-matching-based SDE learning framework, called FlowSDE, to model the non-equilibrium dynamics and identify its critical state transitions. FlowSDE combines conditional flow matching and physical constraints to construct data-driven representations of the system's evolution, capturing mean-field potentials and detecting transition points. We validate FlowSDE with various dynamical systems, demonstrating its ability to uncover latent transition points and predict transitions with greater accuracy. We also compare FlowSDE with traditional early warning signals (e.g. variance, lag-1 autocorrelation (lag-1 AC)) in epileptor models, highlighting the robustness of our approach under non-stationary or noisy conditions. Overall, the FlowSDE framework offers an interpretable, flexible and robust solution to characterize and forecast complex dynamics, with a wide range of potential applications in neuroscience and other fields. This article is part of the theme issue 'Critical transitions and intelligent control in complex systems'.